Sequence Lorentz Spaces and Their Geometric Structure

Abstract

This article is dedicated to geometric structure of the Lorentz and Marcinkiewicz spaces in case of the pure atomic measure. We study complete criteria for order continuity, the Fatou property, strict monotonicity, and strict convexity in the sequence Lorentz spaces γ p,w . Next, we present a full characterization of extreme points of the unit ball in the sequence Lorentz space γ 1,w . We also establish a complete description up to isometry of the dual and predual spaces of the sequence Lorentz spaces γ 1,w written in terms of the Marcinkiewicz spaces. Finally, we show a fundamental application of geometric structure of γ 1,w to one-complemented subspaces of γ 1,w .

Geometry of the Marcinkiewicz Sequence Space

“…Our final section contains a description of the geometry of the unit ball of (m 0 Ψ ) the dual of m 0 Ψ . In particular, we are able to characterise the weak * exposed points of the unit ball of the Lorentz spaces d(w, 1) and γ 1,w extending result of Kamińska, Lee and Lewicki, [13,Theorem 2.6], and of Ciesielski and Lewicki, [8,Theorem 4.7], which characterised the extreme points of d(w, 1) and γ 1,w respectively. We then use our results to give a characterisation of the multipliers of m 0 Ψ .…”

“…We assume that w 1 = 0. Recall that in [8] the sequence Lorentz space, γ 1,w , is defined as all sequences of complex numbers (z n ) n such that…”

“…It is shown in [8,Theorem 5.4] that if ∞ k=1 w k diverges then γ 1,w is the dual of the Marcinkiewicz sequence space m 0 Ψ where Ψ is the symbol given by Ψ(n) = φ γ 1,w (n) for all n.…”

“…Applying Corollary 4.2 we see that the above set is also the 'set of the unit ball of γ 1,w . This provides us with an alternative proof of [8,Theorem 4.7]. Now, we describe the set of weak * -exposed (and extreme points) of the unit ball of (m 0 Ψ ) when (m 0 Ψ ) = 1 , for two renormings of this space.…”